Back in the summer of 2012, emboldened by how nicely the calculation π₁(S¹) had gone, I asked a summer research intern, Joseph Lee, to work on formalizing a proof that the higher-inductive definition of the torus (see Section 6.6 of the HoTT book) … Continue reading →

For those of you who have been waiting with bated breath to find out what happened to your favorite characters after the end of Chapter 8 of the HoTT book, there is now a new installment: Eilenberg-MacLane Spaces in Homotopy Type Theory Dan … Continue reading →

The fact, due to Voevodsky, that univalence implies function extensionality, has always been a bit mysterious to me—the proofs that I have seen have all seemed a bit non-obvious, and I have trouble re-inventing or explaining them. Moreover, there are … Continue reading →

I’m giving a talk this week at CPP. While I’m going to talk more broadly about applications of higher inductive types, for the proceedings, Guillaume Brunerie and I put together an “informalization” of πn(Sn), which you can find here. This is … Continue reading →

A little while ago, we gave an overview of the kinds of results in homotopy theory that we might try to prove in homotopy type theory (such as calculating homotopy groups of spheres), and the basic tools used in our synthetic approach … Continue reading →

Many of us working on homotopy type theory believe that it will be a better framework for doing math, and in particular computer-checked math, than set theory or classical higher-order logic or non-univalent type theory. One reason we believe this … Continue reading →